Let \(G = (V, E)\) a (simple undirected) graph with (finite) set of vectors \(V\) and (finite) set of edges \(E\).
The terms "graph" and "(social) network" are used here interchangeably.
A (finite) family, \(\mathcal{S} = \mathcal{S}(G) = \{S_{1}, S_{2}, \ldots, S_{s}\}\) (of \(s \geq 2\) nonempty (sub)sets of vertices) forms a partition of vertices of \(G\), when
A partition into \(s\) (sub)sets of vertices is also referred to as a \(s\)-partition of graph vertices.
Trivially, for every graph \(G\) with \(|V| \geq 2\), there is at least one partition of its vertices.

Examples of Partitions¶

Structural (endogenous) partitions:
- Communities
- Connected components
Ad hoc (exogenous) partitions:
- Layers (or levels)
- Vertex attributes (or labels or colors)
\(G\) is called a multilayer graph comprising \(\ell \geq 0\) layers (or \(\ell\)-layer graph) when there is a \(\ell\)-partition \(\mathcal{L} = \mathcal{L}(G) = \{L_{1}, L_{2}, \ldots, L_{\ell}\}\) of vertices of \(G\).
Such a partition is called called multilayer partition.

Similarity of Partitions¶

Let \(\mathcal{P} = \{P_{1}, P_{2}, \ldots, P_{p}\}\) and \(\mathcal{Q} = \{Q_{1}, Q_{2}, \ldots, Q_{q}\}\) be two (different) partitions of vertices of graph \(G = (V, E)\).
Then, as far as \(|V| \geq p \times q\), partitions \(\mathcal{P}\) and \(\mathcal{Q}\) are called similar whenever, for all \(i = 1, 2, \cdots, p\) and \(j = 1, 2, \cdots, q\),

If \(G\) accepts partition \(\mathcal{P}\) and \(|V| \geq p^2\), partition \(\mathcal{P}\) is called self--similar whenever, for any \(i = 1, 2, \cdots, p\), the (induced) subgraph \(P_i\) accepts the partition \(\mathcal{P}(Pi)\) too.
In this case, graph \(G\) is called self-similar with respect to partition \(\mathcal{P}\).

3-LAYER GRAPHS

1. Syntetic 3-Layer Graphs in the Triangular Topology
2. Syntetic 3-Layer Graphs in the 2-Path Topology
3. Analytic 3-Layer Graphs in the Triangular Topology
4. Analytic 3-Layer Graphs in the 2-Path Topology
5. Temporal Graphs with 3 Slices
6. A Sierpinski (Fractal) Self-Similar Graph

1. Synthetic 3-Layer Graphs in the Triangular Topology ¶

In [1]:

%matplotlib inline
from syntheticThreeLayerGraph import synthetic_three_level, plot_graph

n1=n2=n3=50
p1=p2=p3=0.01
q1=q2=q3=0.01

G,J,FF,DD,edgeList = synthetic_three_level(n1,n2,n3,p1,p2,p3,q1,q2,q3,no_isolates=False)
pos=plot_graph(G,J,FF,DD,n1,n2,n3,d1=1,d2=10.,d3=0.8,nodesize=50,withlabels=False,edgelist=edgeList,layout=False,b_alpha=0.25)

In [2]:

from threeLayerCommunityParition import create_node_comm_graph, plot_graph

broken_graph,broken_partition,npartition = create_node_comm_graph(G,J.nodes(),FF.nodes(),DD.nodes())
plot_graph(G,broken_graph,broken_partition,npartition,J.nodes(),FF.nodes(),DD.nodes(),d1=1.4,d2=5.,d3=0.8,withlabels=False,nodesize=100,layout=False)

In [3]:

from threeLayerConnComponentsPartition import create_node_conncomp_graph, plot_graph

broken_graph,broken_partition,npartition = create_node_conncomp_graph(G,J.nodes(),FF.nodes(),DD.nodes())
plot_graph(G,broken_graph,broken_partition,npartition,J.nodes(),FF.nodes(),DD.nodes(),d1=1.4,d2=5.,d3=0.8,withlabels=False,nodesize=100,layout=False)

In [4]:

from threeLayer3AttributesPartition import create_node_3attri_graph, plot_graph

r1=r2=r3=0.333

broken_graph,broken_partition,npartition = create_node_3attri_graph(G,J.nodes(),FF.nodes(),DD.nodes(),r1,r2,r3)
plot_graph(G,broken_graph,broken_partition,npartition,J.nodes(),FF.nodes(),DD.nodes(),d1=1.4,d2=5.,d3=0.8,withlabels=False,nodesize=100,layout=False)

2. Synthetic 3-Layer Graphs in the 2-Path Topology ¶

In [5]:

from syntheticThreeLayerGraph_l import synthetic_three_level, plot_graph

n1=n2=n3=50
p1=p2=p3=0.01
q1=q2=q3=0.01

G,J,FF,DD,edgeList = synthetic_three_level(n1,n2,n3,p1,p2,p3,q1,q2,q3,no_isolates=False)
pos=plot_graph(G,J,FF,DD,n1,n2,n3,d1=2,d2=3.,nodesize=50,withlabels=False,edgelist=edgeList,layout=False,b_alpha=0.25)

In [6]:

from threeLayerCommunityParition import create_node_comm_graph, plot_graph_stack

broken_graph,broken_partition,npartition = create_node_comm_graph(G,J.nodes(),FF.nodes(),DD.nodes())
plot_graph_stack(G,broken_graph,broken_partition,npartition,J.nodes(),FF.nodes(),DD.nodes(),d1=1.4,d2=5.,d3=0.8,withlabels=False,nodesize=100,layout=False)

In [7]:

from threeLayerConnComponentsPartition import create_node_conncomp_graph, plot_graph_stack

broken_graph,broken_partition,npartition = create_node_conncomp_graph(G,J.nodes(),FF.nodes(),DD.nodes())
plot_graph_stack(G,broken_graph,broken_partition,npartition,J.nodes(),FF.nodes(),DD.nodes(),d1=1.4,d2=5.,d3=0.8,withlabels=False,nodesize=100,layout=False)

In [8]:

from threeLayer3AttributesPartition import create_node_3attri_graph, plot_graph_stack

r1=r2=r3=0.333

broken_graph,broken_partition,npartition = create_node_3attri_graph(G,J.nodes(),FF.nodes(),DD.nodes(),r1,r2,r3)
plot_graph_stack(G,broken_graph,broken_partition,npartition,J.nodes(),FF.nodes(),DD.nodes(),d1=1.4,d2=5.,d3=0.8,withlabels=False,nodesize=100,layout=False)

3. Analytic 3-Layer Graphs in the Triangular Topology ¶

In [9]:

from analyticThreeLayerGraph import analyticThreeLayerGraph, plot_graph

n = 150
p = 0.1
r1 = r2 = r3 = 0.333

G, layer1, layer2, layer3, edgeList = analyticThreeLayerGraph(n,p,r1,r2,r3,G_isolates=True)
plot_graph(G,layer1,layer2,layer3,d1=1.5,d2=5.,d3=0.8,nodesize=100,withlabels=False,edgelist=edgeList,layout=False,alpha=0.07)

In [10]:

from threeLayerCommunityParition import create_node_comm_graph, plot_graph

broken_graph,broken_partition,npartition = create_node_comm_graph(G,layer1,layer2,layer3)
plot_graph(G,broken_graph,broken_partition,npartition,layer1,layer2,layer3,d1=1.4,d2=5.,d3=0.8,withlabels=False,nodesize=100,layout=False)

In [11]:

from threeLayerConnComponentsPartition import create_node_conncomp_graph, plot_graph

broken_graph,broken_partition,npartition = create_node_conncomp_graph(G,layer1,layer2,layer3)
plot_graph(G,broken_graph,broken_partition,npartition,layer1,layer2,layer3,d1=1.4,d2=5.,d3=0.8,withlabels=False,nodesize=100,layout=False)

In [12]:

from threeLayer3AttributesPartition import create_node_3attri_graph, plot_graph

r1=r2=r3=0.333

broken_graph,broken_partition,npartition = create_node_3attri_graph(G,layer1,layer2,layer3,r1,r2,r3)
plot_graph(G,broken_graph,broken_partition,npartition,layer1,layer2,layer3,d1=1.4,d2=5.,d3=0.8,withlabels=False,nodesize=100,layout=False)

4. Analytic 3-Layer Graphs in the 2-Path Topology ¶

In [13]:

from analyticThreeLayerGraph_l import analyticThreeLayerGraph, plot_graph

n = 150
p = 0.01
r1 = 0.333
r2 = 0.333
r3 = 0.333

G, layer1, layer2, layer3, edgeList = analyticThreeLayerGraph(n,p,r1,r2,r3,G_isolates=True)
plot_graph(G,layer1,layer2,layer3,d1=3.5,d2=5.,d3=0.8,nodesize=100,withlabels=False,edgelist=edgeList,layout=False,alpha=0.2)

In [14]:

from threeLayerCommunityParition import create_node_comm_graph, plot_graph_stack

broken_graph,broken_partition,npartition = create_node_comm_graph(G,J.nodes(),FF.nodes(),DD.nodes())
plot_graph_stack(G,broken_graph,broken_partition,npartition,J.nodes(),FF.nodes(),DD.nodes(),d1=1.4,d2=5.,d3=0.8,withlabels=False,nodesize=100,layout=False)

In [15]:

from threeLayerConnComponentsPartition import create_node_conncomp_graph, plot_graph_stack

broken_graph,broken_partition,npartition = create_node_conncomp_graph(G,J.nodes(),FF.nodes(),DD.nodes())
plot_graph_stack(G,broken_graph,broken_partition,npartition,J.nodes(),FF.nodes(),DD.nodes(),d1=1.4,d2=5.,d3=0.8,withlabels=False,nodesize=100,layout=False)

In [16]:

from threeLayer3AttributesPartition import create_node_3attri_graph, plot_graph_stack

r1=r2=r3=0.333

broken_graph,broken_partition,npartition = create_node_3attri_graph(G,J.nodes(),FF.nodes(),DD.nodes(),r1,r2,r3)
plot_graph_stack(G,broken_graph,broken_partition,npartition,J.nodes(),FF.nodes(),DD.nodes(),d1=1.4,d2=5.,d3=0.8,withlabels=False,nodesize=100,layout=False)

5. Temporal Graph with 3 Slices ¶

In [17]:

%matplotlib inline
from syntheticThreeLayerGraph_time import synthetic_three_level, plot_graph

p1=p2=p3=0.1
n=50
G,J,FF,DD,JFD,edgeList = synthetic_three_level(n,p1,p2,p3,J_isolates=True,F_isolates=True,D_isolates=True)
#print edgeList
created_pos=plot_graph(n,G,J,FF,DD,JFD,d1=2.,d2=3.,nodesize=50,withlabels=False,edgelist=edgeList,layout=False,b_alpha=0.05) #d=0.5 #d

MULTILAYERITY WITHIN MULTILAYERITY?¶

Similarity among Structural and Ad Hoc Partitions¶

By Moses Boudourides & Sergios Lenis¶

Sunbelt 2015¶

The Python scripts for all the computations displayed here are available at:¶

https://github.com/mboudour/GraphMultilayerity ¶

Basic Definitions¶

Examples of Partitions¶

Similarity of Partitions¶

1. Synthetic 3-Layer Graphs in the Triangular Topology ¶

2. Synthetic 3-Layer Graphs in the 2-Path Topology ¶

3. Analytic 3-Layer Graphs in the Triangular Topology ¶

4. Analytic 3-Layer Graphs in the 2-Path Topology ¶

5. Temporal Graph with 3 Slices ¶

6. A Sierpinski (Fractal) Self-Similar Graph ¶

MULTILAYERITY WITHIN MULTILAYERITY?¶

Similarity among Structural and Ad Hoc Partitions¶

By Moses Boudourides & Sergios Lenis¶

Sunbelt 2015¶

The Python scripts for all the computations displayed here are available at:¶

https://github.com/mboudour/GraphMultilayerity¶

Basic Definitions¶

Examples of Partitions¶

Similarity of Partitions¶

1. Synthetic 3-Layer Graphs in the Triangular Topology¶

2. Synthetic 3-Layer Graphs in the 2-Path Topology¶

3. Analytic 3-Layer Graphs in the Triangular Topology¶

4. Analytic 3-Layer Graphs in the 2-Path Topology¶

5. Temporal Graph with 3 Slices¶

6. A Sierpinski (Fractal) Self-Similar Graph¶

https://github.com/mboudour/GraphMultilayerity ¶

1. Synthetic 3-Layer Graphs in the Triangular Topology ¶

2. Synthetic 3-Layer Graphs in the 2-Path Topology ¶

3. Analytic 3-Layer Graphs in the Triangular Topology ¶

4. Analytic 3-Layer Graphs in the 2-Path Topology ¶

5. Temporal Graph with 3 Slices ¶

6. A Sierpinski (Fractal) Self-Similar Graph ¶